Third sound and stability of ³He-⁴He mixture films

In this paper, we study sound in ³He-⁴He mixture films on several representative substrates: Nuclepore, glass, Li, and Na. This work extends previous more technical work which focused on the strongly binding substrates Nuclepore and glass. We proceed from a first-principles, microscopic theory utilizing the variational, hypernetted chain/Euler-Lagrange (HNC-EL) theory as applied to inhomogeneous boson systems. We calculate chemical potentials for both the ⁴He superfluid film and the physisorbed ³He. Numerical density derivatives of the chemical potentials lead to the sought-after third sound speeds. On all substrates, the third sound speeds show a series of oscillations driven by the layered structure of the ⁴He film, this is the case even for the very weakly binding Na substrates despite fairly structure-less chemical potentials. We study the effect of the ³He depends sensitively on the particular ⁴He film coverage. The third sound speed can either increase or decrease. In fact, in some regimes, the added ³He destabilizes the film and can drive "layering transitions" leading to quite complicated geometric structures of the film in which the outermost layer consists of phase-separated regimes of ³He and ⁴He. Finally, we examine the range of applicability of the usual film-averaged hydrodynamic description. We find that at least up to film thicknesses of six liquid layers, there is no regime in which the usual hydrodynamic expression is applicable.